Dear Greg,
thanks so much, I think that now I have understood. Please confirm me this 
reading what follows ;-)

To summarize from the beginning, the table I analyzed is the result of a simple 
experiment. Subjects where exposed  to some stimuli 

and they where asked to evaluate the degree of realism of the stimuli on a 7 
point scale (i.e., data in column "response").
Each stimulus was presented in two conditions, "A" and "AH", where AH is the 
condition A plus another thing (let´s call it "H").

Before I wrongly thought that if I do the analysis anova(response ~ 
stimulus*condition) I would have got the comparison between 

the same stimulus in condition A and in condition AH (e.g. stimulus_1_A, 
stimulus_1_AH). 

Instad, apparently, the interaction stimulus:condition means that I find the 
differences between the stimuli keeping fixed the condition!!
If this is true then doing the anova with the interaction stimulus:condition is 
equivalent to do the ONE WAY ANOVA  first on 

the subset where all the conditions are A and then on the subset where all the 
conditions are AH? Right?


So if all before is correct, my final question is: how by means of ANOVA can I 
track the  significative differences between the stimuli 

presented in A and AH  condition whitout passing for the t-test? Indeed my goal 
was to find in one hand if globally the condition
AH bring to better results than condition A, and on the other hand I needed to 
know for which stimuli the condition AH brings
better results than condition A.



Finally, Iam burning with curiosity to know the answers to the following two 
questions:
1-  is there a difference between anova(response ~ stimulus*condition) and 
anova(response ~ condition*stimulus)
concerning the interaction part?

2-  doing the anova(response ~ stimulus + condition) give the same results of 
two ONE WAY ANOVA
anova(response ~ stimulus) and anova(response ~ condition) but the advantage is 
that they are presented together in one single output?



Looking forward to knowing your response!


Best regards



________________________________
From: Greg Snow <Greg.Snow@imail.org>

<r-help@r-project.org>
Sent: Thu, January 6, 2011 6:41:49 PM
Subject: RE: [R] Problem with 2-ways ANOVA interactions

You really need to spend more time with a good aov textbook and probably a
consultant that can explain things to you face to face.  But here is a basic 
explanation to get you pointed in the right direction:

Consider a simple 2x2 example with factors A and B each with 2 levels (1 and 
2).  Draw a 2x2 grid to represent this, there are 4 groups and the theory would 
be that they have means mu11, mu12, mu21, and mu22 (mu12 is for the group with A 
at level 1 and B at level 2, etc.).  


Now you fit the full model with 2 main effects and 1 interaction, if we assume 
treatment contrasts (the default in R, the coefficients/tests will be different 
for different contrasts, but the general idea is the same) then the 
intercept/mean/constant piece will correspond to mu11; the coefficient (only 
seen if treated as lm instead of aov object) for testing A will be (mu21-mu11) 
and for testing B will be (mu12-m11).  


Now the interaction piece gets a bit more complex, it is (mu11 - mu12 - mu21 + 
mu22), this makes a bit more sense if we rearrange it to be one of ( (mu22-mu21) 
- (mu12-mu11) ) or ( (mu22-mu12) - (mu21-mu11) );  it represents the difference 
in the differences, i.e. we find how much going from A1 to A2 changes things 
when B is 1, then we find how much going from A1 to A2 changes things when B is 
2, then we find the difference in these changes, that is the interaction (and if 
it is 0, then the effects of A and B are additive and independent, i.e. the
amount A changes things does not depend on the value of B and vis versa).

So testing the interaction term is asking if how much a change in A affects
things depends on the value of B.

This is very different from comparing mu11 to mu12 (or mu21 to mu22) which is 
what I think you did in the t-test, it is asking a very different question and 
using different base assumptions (ignoring any effect of B, additional data, 
etc.).  Note that your test on condition is very significant, this would be more 
similar to your t-test, but still not match exactly because of the differences.

Now your case is more complicated since stimulus has 7 levels (6 df), so the 
interaction is a combination of 6 different differences of differences, which is 
why you need to spend some time in a good textbook/class to really understand 
what model(s) you are fitting.


-- 
Gregory (Greg) L. Snow Ph.D.
Statistical Data Center
Intermountain Healthcare
greg.snow@imail.org
801.408.8111

> -----Original Message-----
> From: r-help-bounces@r-project.org [mailto:r-help-bounces@r-
> project.org] On Behalf Of Frodo Jedi
> Sent: Wednesday, January 05, 2011 4:10 PM
> To: r-help@r-project.org
> Subject: [R] Problem with 2-ways ANOVA interactions
> 
> Dear All,
> I have a problem in understanding how the interactions of 2 ways ANOVA
> work,
> because I get conflicting results
> from a t-test and an anova. For most of you my problem is very simple I
> am sure.
> 
> I need an help with an example, looking at one table I am analyzing.
> The table
> is in attachment
> and can be imported in R by means of this command:
> scrd<-
> read.table('/Users/luca/Documents/Analisi_passi/Codice_R/Statistics_res
> ults_bump_hole_Audio_Haptic/tables_for_R/table_realism_wood.txt',
>  header=TRUE, colClasse=c('numeric','factor','factor','numeric'))
> 
> 
> This table is the result of a simple experiment. Subjects where exposed
> to some
> stimuli and they where asked to evaluate the degree of realism
> of the stimuli on a 7 point scale (i.e., data in column "response").
> Each stimulus was presented in two conditions, "A" and "AH", where AH
> is the
> condition A plus another thing (let´s call it "H").
> 
> Now, what means exactly in my table the interaction stimulus:condition?
> 
> I think that if I do the analysis anova(response ~ stimulus*condition)
> I will
> get the comparison between
> 
> the same stimulus in condition A and in condition AH. Am I wrong?
> 
> For instance the comparison of stimulus flat_550_W_realism presented in
> condition A with the same stimulus, flat_550_W_realism,
> 
> presented in condition AH.
> 
> The problem is that if I do a t-test between the values of this
> stimulus in the
> A and AH condition I get significative difference,
> while if I do the test with 2-ways ANOVA I don´t get any difference.
> How is this possible?
> 
> Here I put the results analysis
> 
> 
> #Here the result of ANOVA:
> > fit1<- lm(response ~ stimulus + condition + stimulus:condition,
> data=scrd)
> >#EQUIVALE A lm(response ~ stimulus*condition, data=scrd)
> >
> > anova(fit1)
> Analysis of Variance Table
> 
> Response: response
>                     Df Sum Sq Mean Sq F value   Pr(>F)
> stimulus             6  15.05   2.509  1.1000   0.3647
> condition            1  36.51  36.515 16.0089 9.64e-05 ***
> stimulus:condition   6   1.47   0.244  0.1071   0.9955
> Residuals          159 362.67   2.281
> ---
> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> 
> 
> #As you can see the p-value for stimulus:condition is high.
> 
> 
> #Now I do the t-test with the same values of the table concerning the
> stimulus
> presented in A and AH conditions:
> 
> flat_550_W_realism
> =c(3,3,5,3,3,3,3,5,3,3,5,7,5,2,3)
> flat_550_W_realism_AH                   =c(7,4,5,3,6,5,3,5,5,7,2,7,5,
> 5)
> 
> > t.test(flat_550_W_realism,flat_550_W_realism_AH, var.equal=TRUE)
> 
> Two Sample t-test
> 
> data:  flat_550_W_realism and flat_550_W_realism_AH
> t = -2.2361, df = 27, p-value = 0.03381
> alternative hypothesis: true difference in means is not equal to 0
> 95 percent confidence interval:
> -2.29198603 -0.09849016
> sample estimates:
> mean of x mean of y
> 3.733333  4.928571
> 
> 
> #Now we have a significative difference between these two stimuli (p-
> value =
> 0.03381)
> 
> 
> 
> Why I get this beheaviour?
> 
> 
> Moreover, how by means of ANOVA I could track the  significative
> differences
> between the stimuli presented in A and AH  condition
> whitout doing the t-test?
> 
> Please help!
> 
> Thanks in advance
> 
> 
> 



      
	[[alternative HTML version deleted]]

